概述

1. Using SDE and reverse-time SDE to extend to continous domain. 使用SDE及反向SDE，将时间T 推广到 连续域
2. Generalize Score matching and Diffusion to inifinite T. 统一 score matching 和 扩散模型， 并将时间 T 推广到 无限
3. Two SDEs derived from 2 Markov chain to use 两个具体的SDE， 分别来自不同的Markov chain, 其中一个是 DDPM使用的chain.
4. Train like Denoising Score Matching or DDPM. Sliced Score matching also works. 训练方式与 denoising score matching/DDPM 很相似
5. Inference/Generation: 1. General SDE sampler using the corresponding reverse-time SDE. 2. Propose Predictor-Corrector sampler (SDE Solver + MCMC) 生成过程 1. 使用常用的SDE数值解法， 2. 提出 用 score-based MCMC 改进
6. Propose a Deterministic sampler, call “probability flow ordinary DE (ODE)”. 提出确定性采样器来生成， 好处有多
7. Architecture Improvements 结构改进
8. class-conditional generation, image imputation and colorization， 实现可控制生成， 基于又一个SDE。

Unified Framework

The authors proposed a unified framework generalizes score matching NCSN and DDPM

It uses Stochastic Differential Equation(SDE) SDE 中文 and reverse-time SDE (derivation English) to extend discrete T (>1000) to infinite continuous T.

The general form of SDE is: $d\boldsymbol{x} = f(\boldsymbol{x}, t) dt \;\;\;\; + G(\boldsymbol{x}, t) d\boldsymbol{w}$ Compared to the diffusion: $x_t \sim \mathcal{N}(\sqrt{1-\beta_t} x_{t-1}, \;\; \beta_t \boldsymbol{I})$

Two detailed SDEs for the framework

1. Variance Exploding (VE) SDE: $dx = \sqrt{\frac{d [\sigma^2(t)]}{dt}}dw$ , derived from the Markov Chain: $x_i = x_{i-1} + \sqrt{\sigma_i^2 - \sigma_{i-1}^2} z_{i-1}$, where $z_{i-1} \sim \mathcal{N}(0, \boldsymbol{I})$ .
2. Variance Preserving (VP) SDE: $dx = -1/2 \beta(t) x dt + \sqrt{\beta(t)} dw$, derived from DDPM's discrete Markov chain.

Training

Train a time-dependent score-based model Eq 7, similar to denoising score matching and also DDPM

Inference/Generation/Sampling after training

1. Apply general SDE Sampler
2. Propose Predictor-Corrector sampler (SDE Solver + MCMC)

Deterministic sampler

They proposed a Deterministic sampler, called "probability flow ordinary DE (ODE)"

Advantages (v.s. Stochastic sample):

1. Exact likelihood computation (DDPM has its own computation, iDDPM improves the results)
2. Manipulating latent representations, for image editing, such as interpolation, and temperature scaling.
3. Uniquely identifiable encoding
4. Efficient sampling, reduce T>1000 -> T<100.

Architecture of U-Net

In Appendix H

5 different improvents

And Exponential Moving Average

Controllable Generation

By solving a conditional reverse-time SDE.

Tasks: class-conditional generation, image imputation and colorization

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